Abstract

$(X_\alpha: \alpha is a strong chain in ℘(ω 1 )/Fin if and only if X β - X α is finite and X α - X β is uncountable for each $\beta . We show that it is consistent that a strong chain in ℘(ω 1 ) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω 1 ) but no strong chain exists: □ ω 1 is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain